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This paper provides a multiscale analytical study of steady incompressible turbulent flow through a channel of either Couette or pressure-driven Poiseuille type. Mathematically, the paper’s two most novel features are that (1) the analysis begins with an underdetermined singular perturbation problem, namely the Reynolds averaged mean momentum balance equation, and (2) it leads to the existence of an infinite number of length scales. (These two features are probably linked, but the linkage will not be pursued.) The paper develops a credible assumption of a mathematical nature which, when added to the initial underdetermined problem, results in a knowledge of almost the complete layer (scaling) structure of the mean velocity and Reynolds stress profiles. This structure in turn provides a lot of other important information about those profiles. The possibility of almost-logarithmic sections of the mean velocity profile is given special attention. The sense in which the length scales are asymptotically propor...

A free boundary problem due to Nishiura and Ohnishi is solved in one space dimension. That problem was derived, during their study of phase separation phenomena in diblock copolymers, as an asymptotic limit of pattern-forming PDEs generalizing that of Cahn and Hilliard. The free boundary problem in one dimension reduces to a linear system of ODEs for the lengths of the intervals between interfaces. This system also arises in a completely different context as the spatial discretization of a simple heat equation in a medium with periodic properties. (The medium is homogeneous in an important special case.) The initial-value problem for this system is completely solved, and global stability results for stationary solutions (in which the interfaces are regularly spaced) are obtained. Nucleation phenomena are briefly discussed.

The dynamics of a material in two phases is studied in the context of phase-field models based on a Landau–Ginzburg free energy functional. They consist of a system of two nonlinear diffusion equat...

The Dirichlet problem for singularly perturbed elliptic equations of the type \[ \varepsilon ^2 \sum {a_{ij} (x,u,\varepsilon u_x )u_{x_i x_j } } + \varepsilon \sum {a_i (x,u,\varepsilon u_x )u_{x_i } } - u + f(x,\varepsilon ) = 0\] is treated. An asymptotic series for the solution, containing boundary layer and interior terms is developed, and its uniform asymptotic validity proved under certain (not too restrictive) conditions on the coefficients and the boundary data.

Asymptotic analysis of flame structure is carried out for some two-reaction networks of “competition” type. When the heat releases of the two reactions are ordered in the same way as their activation energies, multiplicity of traveling wave solutions is exhibited, in which more than one type of flame can result from the same unburned condition. This multiplicity phenomenon is of a purely kinetic nature, and occurs when there is a crossover temperature at which the two constants of reaction are equal. Special attention is devoted to obtaining explicit solutions as well as stability limits in transitional regions.When the ordering is opposite, the phenomenon of a temperature plateau occurs, in which the flame temperature is little affected by changes in the unburned conditions. Flame structure under this condition is also studied.